Part II. p-orbital physics in optical lattices

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Exploring New States of Matter in the
p-orbital Bands of Optical Lattices
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
C. Wu, D. Bergman, L. Balents, and S. Das Sarma,
cond-mat/0701788. C. Wu, W. V. Liu, J. Moore and
S. Das Sarma, PRL 97, 190406 (2006). W. V. Liu
and C. Wu, PRA 74, 13607 (2006).
University of Maryland, 02/05/2007
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Collaborators
L. Balents UCSB
D. Bergman UCSB
S. Das Sarma Univ. of Maryland
W. V. Liu Univ. of Pittsburg
J. Moore Berkeley
Many thanks to I. Bloch, L. M. Duan, T. L. Ho, T.
Mueller, Z. Nussinov for very helpful
discussions.
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Outline

• Introduction.

- Rapid progress of cold atom physics in optical
lattices.

• New direction orbital physics in high-orbital
  bands pioneering experiments.

• New features of orbital physics in optical
  lattices.

Fermions flat bands and crystallization in
honeycomb lattice.
Bosons novel superfluidity with time-reversal
symmetry breaking (square, triangular lattices).
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Bose-Einstein condensation

• Bosons in magnetic traps dilute and weakly
  interacting systems.

5
New era optical lattices

• New opportunity to study strongly correlated
  systems.
• Interaction effects are tunable by varying laser
  intensity.

t inter-site tunneling U on-site interaction
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Superfluid-Mott insulator transition
Mott insulator
Superfluid
Greiner et al., Nature (2001).
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Noise correlation (time of flight) in
Mott-insulators

• Noise correlation function oscillates at
  reciprocal lattice vectors bunching effect of
  bosons.

Folling et al., Nature 434, 481 (2005) Altman et
al., PRA 70, 13603 (2004).
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Two dimensional superfluid-Mott insulator
transition
I. B. Spielman et al., cond-mat/0606216.
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Fermionic atoms in optical lattices

• Observation of Fermi surface.

Esslinger et al., PRL 9480403 (2005)

• Quantum simulations to the Hubbard model.

e.g. can 2D Hubbard model describe high Tc
cuprates?
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New direction orbital physics in optical lattices

• Great success of cold atom physics

BEC, superfluid-Mott insulator transition,
fermion superfluidity and BEC-BCS crossover

• Next focus resolve NEW aspects of strong
  correlation phenomena which are NOT well
  understood in usual condensed matter systems.

• Orbital physics studying new physics of
  fermions and bosons in high-orbital bands.

Good timing pioneering experiments double-well
lattice (NIST) and square lattice (Mainz).
J. J. Sebby-Strabley, et al., PRA 73, 33605
(2006) T. Mueller and I. Bloch et al.
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Orbital physics

• Orbital a degree of freedom independent of
  charge and spin.

• Orbital band degeneracy and spatial anisotropy.

• cf. transition metal oxides (d-orbital bands
  with electrons).

Charge and orbital ordering in La1-xSr1xMnO4
Tokura, et al., science 288, 462, (2000).
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New features of orbital physics in optical
lattices

• px,y-orbital physics using cold atoms.

• Strong anisotropy.

Fermions flat band, novel orbital ordering
Bosons frustrated superfluidity with
translational and time-reversal symmetry breaking

• System preparation

fermions s-band is fully-filled p-orbital bands
are active.
bosons pumping bosons from s to p-orbital bands.
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Double-well optical lattices
J. J. Sebby-Strabley, et al., PRA 73, 33605
(2006).

• Laser beams of in-plane and out-of-plane
  polarizations.

White spotslattice sites. Note the difference in
lattice period!
Combining both polarizations

• The potential barrier height and the tilt of the
  double well can be tuned.

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Transfer bosons to the excited band
Grow the long period lattice
Avoid tunneling (diabatic)
Create the excited state (adiabatic)
Create the short period lattice (diabatic)

• Band mapping.
• Phase incoherence.

M. Anderlini, et al., J. Phys. B 39, S199 (2006).
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Ongoing experiment pumping bosons by Raman
transition

• Long life-time phase coherence.

• Quasi-1d feature in the square lattice.

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Outline

• Introduction.

Orbital physics good timing for studying new
physics of fermions and bosons in high-orbital
bands.

• New features of orbital physics in optical
  lattices.

Fermions flat bands and crystallization in
honeycomb lattice.
Bosons novel superfluidity with time-reversal
symmetry breaking (square, triangular lattices).
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p-orbital fermions in honeycomb lattices
pxy-orbital flat bands interaction effects
dominate.
C. Wu, D. Bergman, L. Balents, and S. Das Sarma,
cond-mat/0701788
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px, py orbital physics why optical lattices?

• pz-orbital band is not a good system for orbital
  physics.

isotropic within 2D non-degenerate.

• Interesting orbital physics in the px,
  py-orbital bands.

1/r-like potential

• However, in graphene, 2px and 2py are close to
  2s, thus strong hybridization occurs.

• In optical lattices, px and py-orbital bands are
  well separated from s.

p
s
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Artificial graphene in optical lattices

• Band Hamiltonian (s-bonding) for spin- polarized
  fermions.

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Flat bands in the entire Brillouin zone!

• If p-bonding is included, the flat bands acquire
  small width at the order of .

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Enhance interactions among polarized fermions

• Hubbard-type interaction

• Problem contact interaction vanishes for
  spinless fermions.

• Use fermions with large magnetic moments.

• Under strong 2D confinement, U is repulsive and
  can reach the order of recoil energy.

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Exact solution with repulsive interactions!

• Crystallization with only on-site interaction!

• Closest packed hexagons avoiding repulsion.

• The crystalline order is stable even with
  if .

• The result is also good for bosons.

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Orbital ordering with strong repulsions

• Various orbital ordering insulating states at
  commensurate fillings.

• Dimerization at ltngt1/2! Each dimer is an
  entangled state of empty and occupied states.

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Experimental detection

• Transport tilt the lattice and measure the
  excitation gap.

• Noise correlations of the time of flight image.

G reciprocal lattice vector for the enlarged
unit cells for bosons, - for fermions.
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Open problems exotic states in flat bands

• Divergence of density of states.

• Interaction effects dominate due to the
  quenched kinetic energy cf. fractional quantum
  Hall physics.

• A realistic system for flat band ferromagnetism
  (fermions with spin).

• Pairing instability in flat bands. BEC-BCS
  crossover? Is there the BCS limit?

• Bosons in flat-bands highly frustrated system.
  Where to condense? Can they condense? Possible
  Bose metal phase?

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Outline

• Introduction.

• New features of orbital physics in optical
  lattices.

Fermions flat bands in honeycomb lattice.
Bosons novel superfluidity with time-reversal
symmetry breaking.
W. V. Liu and C. Wu, PRA 74, 13607 (2006) C.
Wu, W. V. Liu, J. Moore and S. Das Sarma, PRL 97,
190406 (2006).
Others related work V. W. Scarola et. al, PRL,
2005 A. Isacsson et. al., PRA 2005 A. B.
Kuklov, PRL 97, 2006 C. Xu et al.,
cond-mat/0611620 .
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Main results superfluidity of bosons with time
reversal symmetry breaking

• On-site orbital angular momentum moment (OAM).

• Square lattice staggered OAM order.

• Triangular lattice stripe OAM order.

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On-site interaction in the p-band orbital
Hunds rule

• Ferro-orbital interaction Lz is maximized.

• cf. Hunds rule for electrons to occupy
  degenerate atomic shells total spin is maximized.

• cf. pip pairing states of fermions 3He-A,
  Sr2RuO4.

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Band structure 2D square lattice

• Anisotropic hopping and odd parity

s-bond
p-bond

• Band minima Kx(p,0), Ky(0,p).

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Superfluidity with time-reversal symmetry
breaking

• Interaction selects condensate as

• Time-reversal symmetry breaking staggered
  orbital angular momentum order.

• Time of flight (zero temperature) 2D coherence
  peaks located at

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Quasi-1D behavior at finite temperatures

• Because , px-particles can maintain
  phase coherence within the same row, but loose
  phase inter-row coherence at finite temperatures.

• Similar behavior also occurs for py-particles.

• The system effectively becomes 1D- like as shown
  in the time of flight experiment.

A. Isacsson et. al., PRA 72, 53604, 2005
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Band structure triangular lattice
CW, W. V. Liu, J. Moore, and S. Das Sarma, Phys.
Rev. Lett. (2006).
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Novel quantum stripe ordering

• Interactions select the condensate as (weak
  coupling analysis).

• Time-reversal, translational, rotational
  symmetries are broken.

• cf. Charge stripe ordering in solid state
  systems with long range Coulomb interactions.
  (e.g. high Tc cuprates, quantum Hall systems).

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Stripe ordering throughout all the coupling
regimes

• Orbital configuration in each site

weak coupling
strong coupling

• cf. Strong coupling results also apply to the
  pip Josephson junction array systems ( e.g.
  Sr2RuO4).

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Time of flight signature

• Stripe ordering even persists into
  Mott-insulating states without phase coherence.

• Predicted time of flight density distribution
  for the stripe-ordered superfluid.

• Coherence peaks occur at non-zero wavevectors.

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Summary

• Good timing to study orbital physics in optical
  lattices.

• New features novel orbital ordering in flat
  bands
• novel superfluidity breaking time reversal
  symmetry.

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Strong coupling vortex configuration of in
optical lattices
Ref C. Wu et al., Phys. Rev. A 69, 43609 (2004).
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Hidden Symmetry and Quantum Phases in Spin 3/2
Cold Atomic Systems
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
Ref C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev.
Lett. 91, 186402(2003) C. Wu, Phys. Rev.
Lett. 95, 266404 (2005) S. Chen, C. Wu,
S. C. Zhang and Y. P. Wang, Phys. Rev. B 72,
214428 (2005) C. Wu, J. P. Hu, and S. C.
Zhang, cond-mat/0512602. Review paper C.
Wu, Mod. Phys. Lett. B 20, 1707 (2006).
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Phase stability analysis
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Px,y-band structure in triangular lattices
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Strong coupling analysis

• Each site is characterized by a U(1) phase ,
  and an Ising variable .

• Inter-site Josephson coupling effective vector
  potential.

J. Moore and D. H. Lee, PRB, 2004.
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Bond current
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Strong coupling analysis

• The minimum of the effective flux per plaquette
  is .

• The stripe pattern minimizes the ground state
  vorticity.

• cf. The same analysis also applies to pip
  Josephson junction array.

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Double well ? triangular lattice
frustration
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Condensation occurs at
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